## On static three-manifolds with positive scalar curvature.(English)Zbl 1385.53020

Summary: We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein $$(n + 1)$$-manifold associated to a maximal domain of a static $$n$$-manifold where the static potential is positive. There are examples where this construction inevitably produces an Einstein metric with conical singularities along a codimension-two submanifold. By proving versions of classical results for Einstein four-manifolds for the singular spaces thus obtained, we deduce some classification results for compact static three-manifolds with positive scalar curvature.

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

### Keywords:

Bochner-type formula; Einstein metric; singular spaces
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